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“Dispersion trading: an empirical analysis on the S&P 100 options”
AUTH ORS
Pierpaolo Ferrari
Gabriele Poy
Guido Abate
ARTICLE INFO
Pierpaolo Ferrari, Gabriele Poy and Guido Abate (2019). Dispersion trading: an
empirical analysis on the S&P 100 options. Investment Management and
Financial Innovations, 16(1), 178-188. doi:10.21511/imfi.16(1).2019.14
DOI http://dx.doi.org/10.21511/imfi.16(1).2019.14
RELEASED ON Wednesday, 06 March 2019
RECE IVED ON Wednesday, 16 January 2019
ACCEPTED ON Thursday, 07 February 2019
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ISSN PRINT 1810-4967
ISSN ONLINE 1812-9358
PUBLISHER LLC “Consulting Publishing Company “Business Perspectives”
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© The author(s) 2019. This publication is an open access article.
businessperspectives.org
178
Investment Management and Financial Innovations, Volume 16, Issue 1, 2019
http://dx.doi.org/10.21511/im.16(1).2019.14
Abstract
is study provides an empirical analysis back-testing the implementation of a
dispersion trading strategy to verify its protability. Dispersion trading is an arbitrage-
like technique based on the exploitation of the overpricing of index options, especially
index puts, relative to individual stock options. e reasons behind this phenomenon
have been traced in literature to the correlation risk premium hypothesis (i.e., the hedge
of correlations dris during market crises) and the market ineciency hypothesis.
is study is aimed at evaluating whether dispersion trading can be implemented with
success, with a focus on the Standard & Poor’s 100 options. e risk adjusted return
of the strategy used in this empirical analysis has beaten a buy-and-hold alternative
on the S&P 100 index, providing a signicant over-performance and a low correlation
with the stock market. e ndings, therefore, provide an evidence of ineciency in
the US options market and the presence of a form of “free lunch” available to traders
focusing on options mispricing.
Pierpaolo Ferrari (Italy), Gabriele Poy (Italy), Guido Abate (Italy)
BUSINESS PERSPECTIVES
LLC “P “Business Perspectives”
Hryhorii Skovoroda lane, 10, Sumy,
40022, Ukraine
www.businessperspectives.org
Dispersion trading:
an empirical analysis
on the S&P 100 options
Received on: 16 of January, 2019
Accepted on: 7 of February, 2019
INTRODUCTION
Traditionally, volatility has been regarded as a measure of risk in port-
folio management; however, since the seminal works by Black and
Scholes (1973) and Merton (1973), investment strategies focused on
volatility have been the object of study both by scholars and practi-
tioners. e presence of liquid derivatives markets and the availabil-
ity of seemingly unlimited computing power have made possible the
practical implementation and testing of complex quantitative ap-
proaches and the development of a whole new category of investment
techniques: volatility trading (Sinclair, 2013). ese techniques aim at
taking prot, regardless of price movements, from the increment or
reduction of volatility of listed securities by making use of derivatives.
One of these techniques is dispersion trading, dened as the practice
of selling index volatility while buying the volatility of its constituents
at the same time (Ren, 2010). Plain vanilla options are the instruments
most widely used by dispersion traders, but more complex and exotic
derivatives, such as variance swaps, can be used for this trading strat-
egy (Hilpisch, 2017).
Due to the absence of empirical analyses of the practical viability of
dispersion trading during the last decade, it is an open issue whether
it can be still used with success. As a consequence, the implementation
of dispersion trading requires a deeper investigation and this article
is aimed at evaluating it in a realistic framework, using more recent
data than the ones available in literature. In particular, this empirical
analysis is focused on the US derivatives market.
© Pierpaolo Ferrari, Gabriele Poy,
Guido Abate, 2019
Pierpaolo Ferrari, Aliate Professor
of Financial Markets and Institutions,
Banking and Insurance Department,
SDA Bocconi School of Management,
Milan, Italy; Full Professor of
Financial Markets and Institutions,
Department of Economics and
Management, University of Brescia,
Ital y.
Gabriele Poy, Research Fellow at
Banking and Insurance Department,
SDA Bocconi School of Management,
Milan, Italy.
Guido Abate, Assistant Professor of
Financial Markets and Institutions,
Department of Economics and
Management, University of Brescia,
Ital y.
volatility trading, options arbitrage, correlation risk
premium, options market ineciency
Keywords
JEL Classification G11, G12
is is an Open Access article,
distributed under the terms of the
Creative Commons Attribution 4.0
International license, which permits
unrestricted re-use, distribution,
and reproduction in any medium,
provided the original work is properly
cited.
179
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e article is organized as follows. e rst section describes dispersion trading, taking into account
the factors underlying its rationale and the relevant literature. Its rst sub-section is devoted to the
theoretical foundations of this technique, while the second sub-section illustrates the technical
problems encountered in the practical implementation of this trading strategy according to the relevant
literature. e second section provides an empirical analysis and is divided into three sub-sections: the
rst one denes the time series of daily market data; the second provides a step-by-step description
of the methodology followed in order to implement a simulation of dispersion trading in a realistic
environment; the third discusses the results. Conclusions are outlined in the last section.
1. LITERATURE REVIEW
An in-depth analysis of dispersion trading re-
quires a review of the available literature, taking
into account both its theoretical and technical
aspects.
1.1. Theoretical foundations
Dispersion trading is an arbitrage-like technique,
which is based on the exploitation of the overpric-
ing of index options, especially index puts, rela-
tive to individual stock options. is mispricing
in the options market has been empirically prov-
en in several past studies, among which we recall
Bakshi and Kapadia (2003), Bollen and Whaley
(2004), Dennis et al. (2006), and Driessen et al.
(2009), and its causes have been traced back to an
overestimation of index volatility with respect to
the volatilities of its constituents.
These analyses compare the differences be-
tween implied volatility and sample volatility
of indices and stocks to verify if there is inco-
herence in the pricing of index and stock op-
tions. Despite their different methodological
approaches and samples (the Standard & Poor’s
500 or the Standard & Poor’s 100), these stud-
ies reach uniform conclusions. They show that
the difference between implied and sample vari-
ances is larger for index options than for stock
options. Driessen et al. (2009) have measured
the implied volatility on the S&P 100 options
to be higher than the sample variance by about
3.89% annually between 1996 and 2003, and by
2.47% on single stocks. In comparison, Bakshi
and Kapadia (2003) covered the period between
1991 and 1995 and found an implied volatility
on stock options in excess of about 1% per year
compared to the sample one, significantly lower
than the 3.3% measured on the S&P 500. Bollen
and Whaley (2004) confirmed these findings
and reported a more pronounced volatility
skew for the S&P 500 options when compared
to stock options; therefore, implied volatility is
less stable across maturities for index options
than for the options written on its constituents.
e variance of returns of an index can be esti-
mated as follows:
2 22
,
11
2 ,
n nn
I i i i j i j ij
i i ji
w ww
= = >
= +
∑ ∑∑
σ σ σσ ρ
(1)
where
i
w
and
i
σ
are the weight of the i-th stock
in the index and its standard deviation, respec-
tively, and
,ij
ρ
is the correlation between the i-th
and j-th constituents.
As a consequence, the variance of the index de-
pends not only upon the variances of its constitu-
ents, but also their correlations. e same relation-
ship in equation (1) theoretically applies not only
for the sample estimates, but also for the variances
implied in index options and the options written
on its constituents. As noted, this is not empiri-
cally proven and, on the contrary, the overesti-
mation of the implied variance of index options
can be traced back to an excess implied correla-
tion among its constituents. Implied correlation,
which is in a way a mean of all the possible corre-
lations between couples of stocks in an index, can
be calculated as follows:
2 22
1
1
,
2
n
I ii
i
imp nn
i ji j
i ji
w
ww
=
= >
−
=
∑
∑∑
σσ
ρσσ
(2)
where I
σ
and
i
σ
are the standard deviation im-
plied in the index options and in the options writ-
ten on its i-th constituent, respectively. Despite the
fact that Pearson’s correlation index is bounded in
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the [−1, +1] interval, this formulation of the index’s
implied correlation can, and oen does, reach val-
ues above +1, stressing the evidence of its overesti-
mation by the options market.
Dispersion trading exploits this mispricing by
opening trades when the dierence between the
implied volatility of the index options and its
theoretical value (i.e. the actual volatility of the
index returns) is at its maximum, aiming at a
convergence between these two measures before
the expiry of the derivative contracts used for
this strategy. is convergence can also be the
result of a reversal of the implied correlation to
its long-term mean. Usually, a dispersion trade
is made of a short position on the index’s implied
volatility by using a short straddle on its options
and a long position on the implied volatility of its
constituents, achieved by opening a long straddle
on the constituents’ options. In other cases, the
implied volatility of index options can be lower
than its theoretical value and, therefore, dispersion
trading is implemented the opposite way, that is,
by going long on an index straddle and shorting
straddles on its constituents.
Deng (2008) has empirically measured a system-
atic and signicant protability of dispersion trad-
ing on the US stock market until 2000, but also a
relevant decrease in the trading results aer this
year. Meanwhile, Marshall (2009) has shown that
a protable use of dispersion trading was also pos-
sible aer the year 2000 by making use of some
simple indicators.
e factors underlying this phenomenon have
been traced in literature to the risk premium and
market ineciency hypotheses.
e rst hypothesis was put forward by Bakshi
and Kapadia (2003), Driessen et al. (2009), and
Bollerslev and Todorov (2011). Based on this hy-
pothesis, the index options’ overpricing in relation
to the options on the index’s constituent stocks is
caused by the presence of a risk premium for the
increase of correlation among these constituents.
Investors diversify their portfolio to reduce risk.
Diversication benets result from the imperfect
correlation of assets and therefore, portfolio vari-
ance is lower than the sum of the variances of its
constituents (subadditivity property).
However, contextual increases in both the volatil-
ity and correlations among stocks during market
downturns are a known and proven phenomenon
(Ang & Chen, 2001). A possible explanation may
be identied in the so-called “panic-selling” that
is derived from a sudden increase in investors’ risk
aversion, which causes a phase-locking and in-
crease of correlations (correlations dri) and con-
versely, a reduction of diversication benets and
an increase in market volatility.
If we apply the usual portfolio variance formula by
using the implied volatilities of index constituents
as inputs as shown below:
2
,
11
,
nn
I i j i j ij
ij
ww
= =
=∑∑
σ σσ ρ
(3)
it becomes apparent that implied volatilities and
prices of index options will be affected by both
the increase of variance in single stocks and
the phase-locking of correlations. As noted by
Driessen et al. (2009), the index options’ over-
valuation is due to an overestimation of correla-
tions among constituents and this excess of im-
plied correlation includes the risk premium for
correlation risk, that is, the risk of increases in
correlations among the index constituents. Said
differently, index options can be regarded as a
form of hedge on this source of risk and, there-
fore, dispersion traders are protection sellers
and their exposure to correlation risk is remu-
nerated by the premium paid by hedgers.
The second hypothesis regarding the source
of index options’ mispricing is based on the
assumption of market inefficiency. If the as-
sumptions of the Black-Scholes-Merton model
were empirically verified, option prices should
not be influenced by demand, but only by fac-
tors linked to the underlying security such as
money market rates and time and therefore, the
no-arbitrage condition should hold (Hull, 2018).
However, Shleifer and Vishny (1997) and Liu
and Longstaff (2004) have discovered the pres-
ence of serious limits in the practical imple-
mentation of arbitrages. For example, losses in-
curred during arbitrages can force the closing of
the operation with a loss before the convergence
of prices to their equilibrium level happens.
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For the same reason, Bollen and Whaley (2004)
stated that market makers are not willing to sell
any amount of the same option at a constant price.
In actuality, when the amount sold increases, the
costs of hedging the exposure to volatility force
the market maker to ask for a higher price for the
option. From this it can be inferred that the supply
curve is positively sloped (Constantinides & Lian,
2018), that is, demand has a direct relationship
with option prices. Gârleanu et al. (2009) have
measured a net positive demand of index options
and a net negative demand for individual stock
options. e causes of this phenomenon are prob-
ably rooted in common investment funds’ policies.
eir passive or semi-passive management styles
require the use of index options for their hedging
strategies, not of derivatives written on individu-
al stocks. Given this pressure on demand and the
positive slope of the supply curve, index options
are systematically overpriced when compared to
stock options.
Deng (2008) provides an in-depth empirical analy-
sis of dispersion trading on the S&P 500 index and
of its underlying factors, that is, risk premium and
market ineciency. e reform of the US options
markets in 1999 (aer an intervention by the SEC
aimed at limiting anti-competitive practices) pro-
vides a “natural experiment” useful in distinguish-
ing the ineciency factor in options pricing, given
the dramatic contraction of transaction costs and
of bid-ask spreads that occurred aer its adoption
(De Fontnouvelle et al., 2003). If overpricing of in-
dex options were a consequence of risk premium
alone, the impact of the 1999 reform on the prof-
its of dispersion trading should be negligible. Deng
(2008) has measured a mean return for this strat-
egy (implemented by buying at-the-money strad-
dles on the S&P 500 and selling straddles on each
of its constituents) of 24% between 1996 and 2000
and –0.03% between 2001 and 2005. is outcome
provides strong evidence of the role of market inef-
ciency, while it is still not entirely possible to rule
out the presence of a limited risk premium.
1.2. Implementation
and technical issues
Non-directional trades, such as dispersion trad-
ing, require a delta-neutral position on the returns
of the underlying index and stocks. Delta hedg-
ing, while appealing on a theoretical perspective,
poses serious issues when applied in practice. First
of all, options’ delta is not constant and rebalanc-
ing is subject to transaction costs. is requires a
reduction of its frequency which, as a consequence,
leads to an imperfect hedge. As mentioned, dis-
persion trading can be implemented by buying
and selling straddles and thus its initial delta is
zero. However, in order to maximize the exposure
to volatility, the options involved in this operation
are at the money, which are characterized by the
highest gamma and, at the same time, the high-
est volatility of the delta. erefore, the ecacy of
delta hedging is limited given the changes of delta
before a new rebalancing, making it a suboptimal
solution.
As previously noted, dispersion trading derives
prot from the overpricing of index options com-
pared to stock options due to an overestimation of
the index’s implied volatility. What makes it possi-
ble to isolate the eects of implied volatility on op-
tion prices is the greek “vega”, that is, the rst par-
tial derivative of the price function with respect
to volatility. erefore, in order to maximize the
return of the strategy, it is necessary to maximize
the exposure of option prices to implied volatility
which is the absolute value of their vega. Moreover,
in the presence of a vega close to zero, the realign-
ment of implied volatilities with their true value
would not have any signicant impact on option
prices and thus on the protability of dispersion
trading. Like delta, vega also varies with the price
of the underlying asset: it is maximum for at-the-
money options and minimum for deep in- or out-
of-the-money options. erefore, in order to keep
vega as distant from zero as possible, it is neces-
sary to rebalance the portfolio towards the at-the-
money options.
Vega plays a crucial role in dispersion trading,
because it is the input for the calculation of
the number of options bought for each index
constituent in order to keep the positive vega on
the portfolio of long straddles on stocks as close
as possible to the absolute value of the negative
vega on short straddles on the index. This way,
the trade is hedged from changes in the implied
volatilities of the individual constituents and
the investment is exposed only to the changes
in the implied volatility of the index resulting
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from an increase or reduction of correlation as
implied in the price of index options.
Theta is seldom taken into account in dispersion
trading because of its marginal, albeit not
irrelevant, role. Given that the passing of time
negatively affects the price of options, the
strategy is best kept as close as possible to a
zero theta, taking into account the positive
theta of the long straddles on stocks and the
negative theta of the short straddles on the
index. Therefore, the calculation of the number
of options to be included in the strategy should
take the form of a joint minimization of the
algebraic sums both of the vegas and the thetas
of the straddles involved.
A method to keep a perfect delta hedge and a
high vega regardless of an option’s remaining
time to expiry and of the price of its underlying
asset is to make use of derivative contracts
known as variance swaps (Nelken, 2006; Härdle
& Silyakova, 2012). A variance swap is a forward
contract that pays the difference between the
realized variance (floating leg) of the underlying
asset and a predefined strike variance (fixed
leg) multiplied by the notional value at maturity.
Unlike options, varia nce swaps do not require the
payment of a premium and unlike plain vanilla
swaps they contemplate only one payment at
expiry. Given their technical features, variance
swaps are comparable to forward contracts and
are also known as realized volatility forwards
(Demeterfi et al., 1999).
Formally, the payo of a variance swap at expiry
is equal to:
( )
2
var
,
R
Payoff K N
σ
=−⋅
(4)
where
2
R
σ
is the variance of the returns of the un-
derlying on an annual basis,
var
K
is the so-called
“delivery price” (strike variance),
N
is the notion-
al value of the variance swap contract.
Given the no-arbitrage condition, the delivery
price of variance swaps can be replicated by an op-
tions portfolio, but unlike these latter derivatives,
variance swaps are not subject to factors dierent
from volatility and time to expiry. erefore, the
rst and clear advantage of using variance swaps
is the absence of the practical problems typical
of delta hedging. Moreover, as noted by Nelken
(2006), variance swaps allow keeping a constant
vega without the need for rebalancing.
Despite these apparent advantages, the use of
variance swaps in dispersion trading poses some
issues. First, the 2008 crisis has caused serious
losses to banks selling this type of derivatives
resulting in a sudden implosion of their market,
especially with regard to contracts written on
single stocks (Carr & Lee, 2009; Martin, 2013)
that are necessary for dispersion trading. Second,
variance swaps are over-the-counter contracts and
thus their market is subject to serious ineciencies
caused by mispricing, transaction costs, and
information asymmetry.
A last alternative technique for the implemen-
tation of dispersion trading exploits the so-
called volatility skew to enhance the return on
this strategy. Implied volatility depends upon
the moneyness of the option as documented in
literature (e.g. Derman & Miller, 2016), and it
decreases with an inverse relationship to strike
prices. This phenomenon implies that mar-
ket operators do not follow the Black-Scholes-
Merton assumption that stock returns are log-
normally distributed; on the contrary, volatility
skew is compatible with a negatively skewed and
leptokurtic distribution. Consequently, disper-
sion trading techniques can take into account
the presence of this deviation from the stan-
dard pricing model and sell strangles instead of
straddles on the index. Specifically, the trader
should sell short at-the-money calls and out
of the money puts which, coherent to volatil-
ity skews, have a higher premium and thus are
more profitable. The trade on the constituent
stocks would be unaltered. This change in the
strategy’s implementation enhances the return
on diversification trading by exploiting the
higher implied volatility of out of the money in-
dex puts.
2. EMPIRICAL ANALYSIS
The following analysis provides a back-testing
of the implementation of a dispersion trading
strategy for the period from January 2010 to
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December 2015 to verify its profitability. The
index selected for this study is the Standard &
Poor’s 100 composed of 100 companies select-
ed from the S&P 500. Companies included in
this index are among the largest and most stable
companies in the S&P 500 and have listed op-
tions (S&P Dow Jones Indices, 2018). The avail-
ability of listed stock and index options is the
primary reason for the choice of the S&P 100
in this study. The options involved are listed
on the most efficient derivative markets world-
wide, causing two contrasting effects. On the
one hand, transaction costs are expected to be
limited; on the other hand, mispricing of index
options is likely to be negligible, negatively af-
fecting the return of dispersion trading.
e strategy is implemented by selling at-the-
money straddles on the S&P 100 and buying
at-the-money straddles on its constituents, but
some enhancements will be applied. Particularly,
a principal components analysis is used to limit
the number of stocks underlying the straddles.
Moreover, an indicator is calculated to signal the
best timing for opening and closing the trades
and, if necessary, to invert the strategy, that is,
to buy straddles on the index and sell straddles
on the constituents. Finally, delta hedging is im-
plemented to guarantee portfolio neutrality in
terms of delta.
2.1. Data sample
An empirical analysis of diversication trading
requires the time series of the options written
both on the S&P 100 and on its constituents and
of the stocks included in the index. As such, for
each listed option with a residual life between
one and 31 days, the following data were extract-
ed from the OptionMetrics database with a daily
frequency:
• identication number;
• closing bid and ask prices;
• expiry date;
• strike price;
• delta;
• vega;
• 30-day implied volatility;
• daily trading volume;
• open interest.
Daily time series data for stocks came from the
Center for Research in Security Prices as follows:
• closing bid and ask prices;
• oating stock;
• dividends;
• splits and reverse splits.
e one-month US Dollar Libor was selected as the
proxy for the risk-free rate and was obtained from
the Federal Reserve Economic Research database.
2.2. Methodology
Options, even on ecient markets, are subject to
transaction costs. One way to limit their impact is
to reduce the number of options bought or sold by
selecting a subsample of constituents that is su-
cient to explain the volatility of the index (i.e., repre-
sentative of the factor structure of the options mar-
ket). Our study makes use of principal components
analysis, in order to reduce the dimensionality of
the problem (Christoersen et al., 2018). is statis-
tical procedure uses an orthogonal transformation
to convert a sample of correlated variables into a set
of linearly uncorrelated ones called principal com-
ponents. In this analysis we adopted the procedure
described in Su (2006), followed also by Deng (2008).
e rst step of this procedure requires the calcu-
lation of the daily continuously compounded re-
turn as follows:
,
,
,1
ln ,
it
it
it
S
rS−
=
(5)
where
,it
S
is the price of the i-th stock at time
.t
e covariance matrix of these returns is estimat-
ed as follows:
( )
2
1,1 1,2 1,
2
**
2,1 2,2 2,
2
,1 ,2 ,
,
k
k
k k kk
Rn
′
= =
σσ σ
σσ σ
σσ σ
RR
V
(6)
where *
R
is the matrix of the centered returns calcu-
lated by subtracting the mean return of each constit-
uent from the corresponding column of the
nk×
matrix of the constituents returns,
n
is the length,
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measured in days, of the time series of returns for
each stock, and
k
is the number of stocks.
Aer dening the inputs, we apply the eigenvalue
decomposition, ordering the eigenvectors of
the covariance matrix, that is, the principal
components, according to the share of variance
explained. In our sample, the rst seven principal
components jointly explain 90.52% of the
sample variance, with the rst and the seventh
components explaining 33.27% and 1.44% of the
total, respectively.
e next step necessary for the practical implemen-
tation of the model is the selection of the stocks that
mimic the seven principal components, based on
the following procedure for each constituent stock:
1) estimate the Pearson correlation coecient of
each of the seven principal components;
2) calculate the weighted arithmetic mean of the
squared correlations using the ratio between
the percentage of variance explained by each
component and the sum of the seven selected
components (for example, for the rst one, the
weight is 33.27%/90.52% = 36.75%)1;
3) order the stocks based on their mean calcu-
lated in step 2 and select the rst 20 stocks in
the ranking (Table 1);
4) regress, without intercept, the daily returns
of the index on those of the 20 selected stocks
(Ta ble 2);
5) discard the stocks, seven in this case, with a
signicance of at least 1%;
6) repeat step 4 for the remaining stocks, 13 in
this case (Table 3).
By applying this procedure, it is possible to iden-
tify the stocks that best explain index volatility
and as a consequence, dispersion trading can be
implemented by buying or selling straddles writ-
ten only on them and not on the 100 constituents
of the selected index.
1 With respect to this step, we depart from Su (2005) wherein the arithmetic mean correlations were not weighted. Our decision is based
on the assumption that it is more useful to select stocks that are highly correlated with the most relevant principal components given the
large dispersion in explained variances.
Table 1. Ranking of stocks
No Ticker Constituent name Weighted mean
correlation
1HON Honeywell International 0.353839326
2MET Metlife 0.348578960
3WFC Wells Fargo 0.333608035
4JPM JPMorgan Chase 0.329813444
5USB US Bancorp 0.324702226
6 BK Bank of New York Mellon 0.3217 10 0 4 9
7CVX Chevron 0.318882532
8SLB Schlumberger 0.316688 635
9EMR Emerson Electric 0.31660 0419
10 BLK Blackrock 0 .31456 0849
11 XOM Exxon 0. 314519303
12 CCitigroup 0.314214965
13 UTX United Technologies 0.310955303
14 CAT Caterpillar 0.309998038
15 MMM 3M 0.308558810
16 OXY Occidental Petroleum 0.307202009
17 GE General Electrics 0.305588500
18 BAC Bank of America 0.292363556
19 RF Regions Financial 0. 284197114
20 AXP American Express 0.280715103
Table 2. Daily returns for 20 stocks
Residuals
Min 1Q Median 3Q Max
–0.0109757 –0.001459 6 0.0001633 0.0016881 0.0120707
Coefficients
Estimate Std. error t-value Pr(> |t|)
HON$return 0.090480 0.009609 9.416 < 2e-16***
MET$return 0.025418 0.006572 3.868 0.000115***
WFC$return 0.042460 0.009078 4.677 3.17e-06***
JPM$return 0.034257 0.008279 4 .138 3.70e-05***
USB$return 0.035095 0.0 09796 3.582 0.000351***
BK$return 0. 011631 0.007444 1.563 0.11837 3
CVX$return 0.045261 0.009928 4.559 5.55e-06***
SLB$return 0.013548 0.006135 2.208 0.027386*
EMR$return 0.014589 0.007825 1.864 0.062468.
BLK$return 0.053779 0.0 06315 8.516 < 2e-16***
XOM$return 0.119519 0.010833 11.03 3 < 2e-16***
C$return 0.018132 0.00610 3 2.971 0.003016**
UTX$return 0.062428 0.0 09637 6.478 1.26e-10***
CAT$return 0.010254 0.006640 1.544 0 .122735
MMM$return 0 .08 4748 0.009544 8.879 < 2e-16***
OXY$return 0.008785 0.0 06718 1. 30 8 0 .1912 25
GE$return 0.06 48 43 0.007 763 8.352 < 2e-16***
BAC$return 0. 0 0 47 17 0.005811 0.812 0.417023
RF$return –0.011055 0.004850 –2.279 0.022783*
AXP$return 0.061899 0.0 06739 9.185 < 2e-16***
Notes: Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1;
residual standard error: 0.002603 on 1,512 degrees of
freedom (1 observation deleted due to missingness); multiple
R-squared: 0.9292; adjusted R-squared: 0.9283; F-statistic:
992.1 on 20 and 1,512 DF, p-value: <2.2e-16.
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Table 3. Daily returns for 13 stocks
Residuals
Min 1Q Median 3Q Max
–0.01065 40 – 0.0 01454 0 0.0001440 0.0016900 0. 0103740
Coefficients
Estimate Std. error t-value Pr(> |t|)
HON$return 0.102563 0.009296 11.03 3 < 2e-16***
MET$return 0.028279 0.006402 4. 417 1.07e-05***
WFC$return 0.042243 0.008918 4.737 2.38e-06***
JPM$return 0.036541 0.007994 4.571 5.25e-06***
USB$return 0.029818 0.009602 3.105 0.0 01935* *
CVX$return 0.058902 0.009407 6.262 4.95e-10***
BLK$return 0.056575 0.006229 9.082 < 2e-16***
XOM$return 0.127405 0.010695 11. 913 < 2e-16***
C$return 0.020944 0.005662 3.699 0.000224***
UTX$return 0.067742 0.009499 7. 13 2 1.53e-12***
MMM$return 0.089705 0.00 9478 9.464 < 2e-16***
GE$return 0.0 69196 0 .007741 8.939 < 2e-16***
AXP$return 0.062760 0.006750 9.298 < 2e-16***
Notes: Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1;
residual standard error: 0.002621 on 1,519 degrees of
freedom (1 observation deleted due to missingness); multiple
R-squared: 0.9279; adjusted R-squared: 0.9273; F-statistic:
1,503 on 13 and 1,519 DF, p-value: <2.2e-16.
An additional requirement for an optimal
implementation of dispersion trading is the
construction of a timing indicator that provides
entry signals for this strategy. As previously
discussed, Deng (2008) has measured a
sharp decrease in performance since the year
2000, but, as noted by Marshall (2009), profit
opportunities have also lasted after that date
provided that trades are opened and closed
following correct timing.
This empirical analysis is primarily based on
the premise that dispersion trading is founded
upon the difference between the correlation
among constituents implicit in index options
and the actual correlation. Therefore, the timing
indicator is calculated, on a daily basis, as the
difference between the implicit correlation of
S&P 100 options and the sample correlation
of the 13 stocks selected, measured on 30-day-
rolling windows. Specifically, the implied
volatility is calculated as the mean between
the implied volatilities of the at-the-money call
and put on each stock. Thereafter, to identify a
proxy of the mean level of correlations among
constituents implied by option prices, equation
(2) is modified as follows:
2 22
,,
1
,,
1
,
2
n
It i it
impl i
tnn
i j it jt
i ji
w
ww
=
= >
−
=
∑
∑∑
σσ
ρσσ
(7)
where ,
It
σ
is the 30-day implied volatility
of index options at time
,t
,it
σ
is the 30-day
implied volatility of stock options on the i-th
constituent at time
,t
i
w
is the weight in the
index of the i-th stock, calculated as in Table 3.
Then, for each day t, the 30-day correlation
between the returns of each couple of the
13 selected constituents is estimated and,
following equation (7), the aggregated measure
of correlation is calculated as follows:
, , ,,
1
,,
1
,
nn
i j it jt i jt
i ji
sam
tnn
i j it jt
i ji
ww
ww
= >
= >
=∑∑
∑∑
σσ ρ
ρσσ
(8)
where
,,i jt
ρ
is the correlation coefficient
between the returns of the i-th and j-th stocks
estimated on the preceding 30 days.
Finally, the timing indicator is calculated as the
difference between (7) and (8):
.
impl sam
tt t
Indicator
ρρ
= −
(9)
When the indicator is high, a long position in
dispersion trading is opened, selling a straddle
on the S&P 100 and buying a straddle on the
subsample of 13 constituents, because, this
way, we invest in a reduction of the implied
correlation, converging toward the sample one.
An opposite position is taken when the indicator
is low, opening a short dispersion trade.
The identification of “high” and “low” levels of
the timing indicator requires the use of Bollinger
bands, calculated as the 30-day moving average
of the indicator plus and minus 1.5 times its
30-day standard deviation (Figure 1). Long and
short dispersion trades are opened when the
indicator breaks above the upper band or breaks
under the lower band, respectively. The list of
the opening dates is provided in Table 4.
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Table 4. List of opening dates
No Date Position
122/03/2010 Long
205/05/2010 Short
313/ 07/2 010 Long
403/12 /2010 Short
506/01/2011 Long
609/ 0 3/2 011 Short
725 / 0 4 /20 11 Long
802 / 0 6 / 2011 Short
923 / 0 9/2011 Long
10 01/12/2011 Short
11 12/ 01/2 012 Long
12 27/ 03 /2012 Short
13 22 /05 /2012 Long
14 26/06/2012 Short
15 26/0 7/2012 Long
16 07/ 09/2012 Short
17 19/ 10/2 012 Long
19 24/12 /2012 Long
20 06/06/2013 Short
21 24/07/ 20 13 Long
22 16/0 9/2 013 Short
23 10/10/2013 Long
24 11/ 10 / 2013 Short
25 20/11/2013 Long
26 19/ 12 /2013 Short
27 11/ 03/ 2 014 Long
28 17/04/ 2014 Short
29 27/05/2014 Long
30 01/08/2014 Short
31 15/ 09/ 2014 Long
32 09/10 /2014 Short
33 19/11/20 14 Long
34 09/01/2015 Short
35 19/ 02 / 20 15 Long
36 01/04/2015 Short
37 29/0 4 /2015 Long
38 11/ 0 6/20 15 Short
39 03/0 8 /2015 Long
40 27/ 08/ 2015 Short
41 15/ 10/2 015 Long
42 07/ 12/2 015 Short
e rst step before the actual implementation
of the back-testing is to verify that every option,
bought or sold, both on the S&P 100 and on its
subsample of constituents, has the same expi-
ry date. e second step is the calculation of the
number of stock options required to be bought or
sold for each index option.
erefore, for each opening date shown in Table 4,
a dispersion trade is opened only if there are cou-
ples of put and call options with the same expiry
and with residual life not less than 10 days. is
additional constraint is included because of the er-
ratic trend typical of options close to expiry.
e number of contracts is calculated to minimize
the dierence between the vega of the S&P 100 op-
tions and the vega of the portfolio of options on
the 13 selected constituents in order to hedge an
increase or decrease in the volatilities of the con-
stituents themselves. e practical implementa-
tion requires the calculation of the vega on S&P
100 options and then the theoretical vega of each
option according to the weight of the i-th stock in
the index
i
w
as follows:
Vega th i ,t=wi ∙ Vega S
&
P100t (10)
Finally, the number of options bought or sold for
each i-th stock is the ratio between the theoretical
vega and the measured vega on the i-th constituent:
Figure 1. Timing indicator and Bollinger bands
-0,6
-0,5
-0,4
-0,3
-0,2
-0,1
0
01.03.2010 01.03.2011 01.03.2012 01.03.2013 01.03.2014 01.03.2015
Indicator Moving average (30 days) Upper band Lower band
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,
,
,
it
it
it
Vega th
Number of options Vega
= (11)
Dispersion trades are then closed when the op-
tions are at seven days from expiry, or when an
opposite trade should be opened according to
the timing indicator.
Moreover, dispersion trading is tested by taking
into account delta hedging given the deltas cal-
culated at market close. The trade is rebalanced
once daily. The delta for the index and each of
its 13 selected constituents is the algebraic sum
of the deltas of calls and puts, taking into ac-
count the long or short position of the trade.
The profit and loss of the delta hedging is cal-
culated as the summation of the products of the
daily variation of the delta and the price (value)
of the corresponding stock (index).
3. RESULTS
e return of the strategy of dispersion trading im-
plemented in this empirical analysis is 23.51% per
year compared to the 9.71% return of the S&P 100
index in the same time span. Given an annual stan-
dard deviation of 9.42% and a risk-free rate of 0.21%,
the Sharpe ratio of the strategy is signicant and
equal to 2.47.
Another interesting feature is the extremely low cor-
relation coecient between the returns of this strat-
egy and those of the S&P 100 equal to a mere 0.0372.
It is clear that dispersion trading is able to provide a
performance completely independent from the stock
market; therefore, with a very limited systematic risk.
ese outcomes are coherent with the ndings on
dispersion trading provided by earlier literature, as
reported in section 1, in particular with Deng (2008)
and Marshall (2009).
CONCLUSION
is article has provided a review of the theoretical basis for dispersion trading and has claried its in-
terpretation as an arbitrage on the mispricing of index options with regard to the overestimation of the
implied correlations among its constituents. e empirical analysis showed how a simple version of dis-
persion trading, implemented using at-the-money plain vanilla straddles on the S&P 100 and a represen-
tative subsample of constituents, has signicantly over performed the stock market, showing almost no
correlation to the chosen index.
While the empirical results show a strong predominance of dispersion trading if compared to a
simple buy-and-hold strategy, a limitation of this analysis is the assumption of the absence of
transaction costs. If we ignore slippage (the difference between the expected price of a trade and
the price at which it is executed), only a market maker could have replicated the performance of
our back-testing. Another limitation of the present analysis is the rather simplified delta hedging
technique based upon a simple daily rebalancing. An optimized hedge would have gained higher
returns, therefore compensating, at least partially, transaction costs. As a consequence, these two
limitations of this study, while present, have opposing effects, and their combined impact is ex-
pected to be negligible, especially given the low number of qualified trades (just 29 in six years) and
their brief mean length (only 16.31 days).
erefore, our analysis provides a strong evidence of ineciencies in the US options market and the
presence of a form of “free lunch” available to traders focusing on options mispricing.
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